Joint measures and cross-covariance operators

Author:
Charles R. Baker

Journal:
Trans. Amer. Math. Soc. **186** (1973), 273-289

MSC:
Primary 60G15; Secondary 28A40

MathSciNet review:
0336795

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Abstract: Let (resp., ) be a real and separable Hilbert space with Borel -field (resp., ), and let be the product measurable space generated by the measurable rectangles. This paper develops relations between probability measures on , i.e., joint measures, and the projections of such measures on and . In particular, the class of all joint Gaussian measures having two specified Gaussian measures as projections is characterized, and conditions are obtained for two joint Gaussian measures to be mutually absolutely continuous. The cross-covariance operator of a joint measure plays a major role in these results and these operators are characterized.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1973-0336795-3

Keywords:
Joint measures,
Gaussian measures,
absolute continuity of measures,
covariance operators,
mutual information

Article copyright:
© Copyright 1973
American Mathematical Society